2005 CMO 第 3 题

As the graph, a pond is divided into 2n (n $\geq$ 5) parts. Two parts are called neighborhood if they have a common side or arc. Thus every part has three neighborhoods. Now there are 4n+1 frogs at the pond. If there are three or more frogs at one part, then three of the frogs of the part will jump to the three neighborhoods repsectively. Prove that for some time later, the frogs at the pond will uniformily distribute. That is, for any part either there are frogs at the part or there are frogs at the each of its neighborhoods.

如图所示，一个池塘被分为 2n (n $\geq$ 5) 个部分。如果两个部分具有共同的边或弧，则称为邻域。因此每个部分都有三个邻域。现在池塘里有 4n+1 只青蛙。如果某一部分有三只或更多青蛙，则该部分的三只青蛙将分别跳到三个邻域。证明一段时间后，池塘里的青蛙会均匀分布。也就是说，对于任何部分，要么该部分有青蛙，要么其每个邻域都有青蛙。